Mathematical Description of NPSC
If a network programming problem with side constraints has
n nodes, a arcs, g nonarc variables, and
k side constraints, then
the formal statement of the problem solved by PROC NETFLOW is

 min {c^{T} x + d^{T} z}
 subject to
 F x = b




 where
 c
is the a x 1 objective function coefficient of arc
variables vector (the cost vector)

x
is the a x 1 arc variable value vector
(the flow vector)

d
is the g x 1 objective function coefficient of
nonarc variables vector

z
is the g x 1 nonarc variable value vector

F
is the n x a nodearc incidence matrix of the
network, where
 F_{i,j} = 1
 if arc j is directed toward node i
 F_{i,j} = 1
 if arc j is directed from node i
 F_{i,j} = 0
 otherwise

b is the n x 1 node supply/demand vector, where
 b_{i} = s
 if node i has supply capability of s units of flow
 b_{i} = d
 if node i has demand d of units of flow
 b_{i} = 0
 if node i is a transshipment node

H is the k x a side constraint
coefficient matrix for arc variables, where
H_{i,j} is the coefficient of arc j in the ith
side constraint

Q
is the k x g side constraint coefficient matrix for
nonarc variables, where
Q_{i,j} is the coefficient of nonarc j in the ith
side constraint

r
is the k x 1 side constraint righthandside vector

l is the a x 1 arc lower flow bound vector

u is the a x 1 arc capacity vector

m is the g x 1 nonarc variable value lower bound
vector

v is the g x 1 nonarc variable
value upper bound vector
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.